An improved construction for spanners of disks

Let D be a set of n pairwise disjoint disks in the plane. Consider the metric space in which the distance between any two disks D and D′ in D is the length of the shortest line segment that connects D and D′. For any real number ε>0, we show how to obtain a (1+ε)-spanner for this metric space that has at most (2π/ε)⋅n edges. The previously best known result is by Bose et al. (2011) [1]. Their (1+ε)-spanner is a variant of the Yao graph and has at most (8π/ε)⋅n edges. Our new spanner is also a variant of the Yao graph.

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